In this article we will discuss about the two main methods used in stability analysis of finite slopes. The methods are: 1. Stability Analysis of Finite Slopes – Culmann’s Method 2. Stability Analysis of Finite Slopes Using the Swedish Circle Method.
1. Stability Analysis of Finite Slopes – Culmann’s Method:
Culmann (1866) considered a plane failure surface and the equilibrium of the wedge of soil above the failure surface is considered by assuming that the inclination of the failure surface will be critical when the stability number is maximum.
Figure 17.6(a) shows the earth slope AB of height, H making an angle of β with the horizontal. Consider a failure surface AE, at some angle of α with the horizontal. From point B, drop a perpendicular BD on the failure surface AE.
The forces acting on the wedge of soil ABEA are – (1) W is the weight of wedge ABEA of soil, acting vertically downwards; (2) C is the cohesive force acting along the surface AE; (3) R is the reaction acting on the surface AE at an angle ɸm with the normal to AC.
We can write –
W = Area of wedge ABEA × γ Þ W = (1/2) × AE × BD × γ
C = cml
where γ is the density of soil, cm is the mobilized cohesion of the soil, ɸm is the mobilized angle of shearing resistance, and l is length AE. Consider ΔABD –
Substituting the value of W from Eq. (17.17) into Eq. (17.18), we have –
The factor of safety for the slope and the stability number can be determined by substituting this value of α, defined by Eq. (17.21) into Eqs. (17.19) and (17.20), respectively.
Investigations carried out in Sweden on slope failures indicated that the failure surface resembles the arc of a circle. Fellenius developed a method for stability analysis of a slope assuming a circular failure surface, which is known as the Swedish circle method.
Stability Analysis for Cohesive Soil (ɸU = 0 Analysis):
Figure 17.8 shows a slope AB, the stability of which is to be determined. A trial slip surface of radius r is assumed and the factor of safety of the slope is determined for the assumed trial slip surface.
Let r be the radius of slip surface about center O, W be the weight of the soil of the wedge ABDA per m length acting through its centroid, and driving or destabilizing moment –
Factor of safety of the slope along trial slip surface AD is –
The distance of the centroid (x̅) of the wedge ABDA from the center of rotation “O” can be determined by dividing the wedge into a number of vertical slices and dividing the algebraic sum of the moment of weight of each slice about centre “O” by the weight of the wedge.
The analysis is repeated for a number of trial slip surfaces and the factor of safety is determined in each case. The slip surface corresponding to minimum factor of safety is the critical slip surface.
c – ɸ Analysis:
For soil, which has both cohesion and friction components of shear strength, that is, c – ɸ soil, the shear strength along the slip surface is also contributed by the frictional component, which is a function of normal stress. The normal stress varies at every point on the slip surface with the horizontal and hence the analysis is done by dividing the wedge ABDA into a number of vertical slices, as shown in Fig. 17.9. The force between the slices is neglected and each slice is assumed to act independently as a column of soil.
The weight Wi of each slice is assumed to act at its center. If this weight is resolved into normal (Ni) and tangential components (Ti, the normal component, Ni will pass through the center of rotation “O” and does not cause any moment. However, the tangential component, Ti, will cause a driving moment MDi = Ti. r, where r is the radius of the slip surface. The tangential components of a few slices at the base may cause resisting moment, in which case Ti for those slices is considered negative.
As per Coulomb’s equation,
Resisting force along the slip surface = (c.Δl + Ni tan ɸ)
Where c is the unit cohesion and Δl is the length of the base of each slice = b. secα
For the entire slip surface AB, driving moment will be –
MD = r. ΣTi
Resisting moment is given by –
MR = r × Resisting force = r(cΣ Δl + tan ɸΣNi)
A number of trial slip surfaces are considered and the factor of safety of the slope for each trial surface is determined. The minimum of the factor of safety so obtained is the factor safety of the slope and the corresponding slip surface is known as critical slip surface.
The Swedish circle method, also known as the method of slices, is a general method, which is equally applicable to homogeneous soils, stratified soils, fully or partially submerged soils, non-uniform soils, and for cases where seepage and pore pressure exist within the soil slope.
Fellenius proposed an empirical procedure to find the center of the most critical slip surface in a pure cohesive soil. For the toe failure case, a point Q can be located by drawing two lines at angles α and Ψ at points A and B, as shown in Fig. 17.10.
The angles α and Ψ depend on the slope as shown in Table 17.1. After obtaining the point Q, the point P is located at a horizontal and vertical coordinates of (4.5H, –H) from the toe. The center of the critical slip surface lies on the extended line of PQ. The line PQ is known as the Fellenius line.
Considering different centers O1, O2, O3, O4, O5, etc., the corresponding trial slip surfaces can be used for toe failure to determine the factor of safety. The factor of safety is then plotted at the corresponding centers normal to the Fellenius line PQ. By joining all these points as shown in Fig. 17.10, the variation of factor of safety can be obtained.
It may be observed from Fig. 17.10 that the radius of slip surface decreases with decrease in distance of the center of slip surface from the slope. Also, the factor of safety is found to decrease significantly with decrease in radius and the curve becomes asymptotic with the Fellenius line nearer the slope.
In the case of base failure, Fellenius showed that for a pure cohesive soil with slope angle less than 53°, the center of critical slip surface lies on a vertical line drawn through the mid-width of the slope and the central angle of the critical slip surface is about 133.5°. When a stiffer layer of soil or rock lies beneath the toe, the critical circle tends to be tangential to this stratum.
The above guidelines can only serve as pointers and in any stability analysis, a sufficiently large number of trials have to be made to locate the most critical slip surface.